This invention relates generally to systems for determining the time-frequency distributions of energy in signals, and more particularly, to systems and apparatus for measuring the energy in signals as a function of frequency and time. The present invention pertains to a signal analysis system which achieves time-frequency distributions, or representations having high resolution with reduced interference from the effects of cross-terms, and which can be realized efficiently in integrated circuitry.
Fourier transformation has long been recognized as a simple and convenient technique for decomposing a signal into its individual frequency components, and obtaining an indication of the intensity of the signal at each such frequency. Conventional Fourier analysis, whether conducted mathematically or in any of several commercially available circuit systems, produces a representation of the frequency distribution of a signal without indicating the time at which the frequencies occurred. Thus, if a signal which has a slowly varying frequency content is Fourier analyzed, the changes in the frequency content of the signal over time would not be evident. One useful and well-known approach to the difficulty discussed above is to analyze the slowly varying signal for small periods of time during which it can be assumed that the signal does not contain rapid changes. In the context of a slowly varying signal, this window concept will provide a useful indication of the variations over time. The duration of the window during which the slowly varying signal can be analyzed can be shortened in response to the rate of change in the frequency content of the signal.
The well-known spectrogram and the short-time Fourier transform are techniques which utilize the window concept, and have become standard techniques in the art. These known systems, however, are not useful in situations where the energy, or spectral, content of the signal varies with such rapidity that the signal cannot reasonably be considered to be stationary for almost any window duration. In this regard, it is to be noted that as the duration of the window is decreased, the frequency resolution of the system is also decreased.
The shortcomings of the spectrogram and the short-time Fourier transform have not prevented these techniques and their variations from becoming essential tools in the technology of signal analysis, but have prompted other work in the development of distributions which are joint functions of time and frequency. The thrust of that effort is to generate a description of the energy density, or intensity, of the signals to be analyzed simultaneously in frequency and time. Clearly, the resulting representation would be a multi-variable representation, and would be of significant benefit in the devising of signals which exhibit desirable time-frequency properties.
Numerous individuals in the prior art have studied the difficulties associated with achieving a simple, yet effective, time-frequency distribution system, one of the most notable being that which has been named after Wigner. The various distributions which have been proposed in the literature have been shown to behave in ways which are dramatically different from one another. Although many of the known distributions have characteristics which satisfy generally accepted criteria, the differences in their behavior have impeded development of a consistent theory. The lack of a consistent theory, however, did not impede development of a generalized analytical approach from which an infinite number of distributions can be derived. The generalized expression is as follows: ##EQU1## In the above expression, the underlined portion has been termed the "kernel" and it is recognized that the character of the distribution may be changed by changing the kernel.
The foregoing notwithstanding, it is now recognized that signals of practical interest often do not conform to the requirements of realistic application of Fourier principles. The Fourier approach works best when the signal of interest is composed of a number of discrete frequency components so that time is not a specific issue. Such would be the case, for example, in a constant frequency sinusoid, or somewhat paradoxically, when the signal exists for a very short period of time so that its time of occurrence is considered to be known. An impulse function is such a signal. Nevertheless, with respect to signals which typically are encountered in practical applications, most such signals cannot be satisfactorily represented, and generally have been considered to be suspect, requiring them either to be forced into the mold, or abandoned.
It has been quite difficult to handle non-stationary signals, such as chirps, using conceptualizations based on stationarity. As indicated, the spectrogram represents an effort to apply the Fourier transform for a short-time analysis window, within which it is hoped that the signal behaves reasonably within the requirements of stationarity. Moving the analysis window in time along the signal, one hopes to track and capture the variations of the signal spectrum as a function of time. If the analysis window is made short enough to capture rapid changes in the signal, it becomes impossible to resolve frequency components which are close in frequency during the analysis window duration.
The well-known Wigner distribution has many important and interesting properties, and provides a high resolution representation in time and in frequency for a non-stationary signal, such as a chirp. In addition, this distribution has the important property of satisfying desired time and frequency marginals. The term "marginal" is obtained from probability theory to indicate an individual distribution. The marginals are derived from the joint distribution by integrating out the other variables.
The Wigner distribution suffers from significant disadvantages. For example, its energy distribution is not non-negative and it often is characterized with severe cross terms, or interference terms, between components in different time-frequency regions. This leads to serious confusion and misinterpretation. The absence of negativity of the energy value is a highly desirable property, which is present in the spectrogram, but not the Wigner distribution. The property of being characterized with proper time and frequency marginals is also highly desirable, and such is the case with the Wigner distribution, but not the spectrogram.
After investigation by many persons of skill in the art, the principle faults of negativity and cross terms have been considered to be facts of life which only can be dealt with by performing additional computations. There is a need for a system for determining time-frequency distributions which have high resolutions in time and frequency and which have considerably reduced cross, or interference terms.
The Choi-Williams distribution was one such attempt. The Choi-Williams distribution, however, requires an enormous amount of computation in practical implementation. The present invention provides a new time-frequency distribution that has a good time-frequency behavior, and a fast computational algorithm as well.